We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a proof for its convergence. The result implies that there are cases of convergence of finite difference schemes applied to ill-posed Cauchy problems.
"Semi-discrete finite difference schemes for the nonlinear Cauchy problems of the normal form." Proc. Japan Acad. Ser. A Math. Sci. 93 (9) 99 - 104, November 2017. https://doi.org/10.3792/pjaa.93.99